Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

I3M, Université Montpellier 2, Place E. Bataillon, 34095 Montpellier, Cedex, France; pierre.pudlo@univ-montp2.fr

Received: 10 July 2008
Revised: 3 March 2009
Revised: 15 June 2009

Abstract

To establish lists of words with unexpected frequencies in long sequences, for instance in a molecular biology context, one needs to quantify the exceptionality of families of word frequencies in random sequences. To this aim, we study large deviation probabilities of multidimensional word counts for Markov and hidden Markov models. More specifically, we compute local Edgeworth expansions of arbitrary degrees for multivariate partial sums of lattice valued functionals of finite Markov chains. This yields sharp approximations of the associated large deviation probabilities. We also provide detailed simulations. These exhibit in particular previously unreported periodic oscillations, for which we provide theoretical explanations.